I watched the video and thought it only explained a very specific case of AR.
A stateful firewall enforces the TCP protocol rules. If, for example, I want my internal users to browse to the Internet, but no one from the Internet can connect to my internal systems, I can use a stateful firewall to enforce the rule that when packet is received from the Internet, it must match an already existing connection. That is, the internal and external devices have already established a TCP connection by following the protocol rules known as the "3-way handshake." Further, the firewall doesn't allow a packet with the SYN flag set to enter from the Internet (because I don't want connections originating from the Internet).
The firewall enforces these rules by monitoring the "state" of each data flow.
The problem of asymmetrical routing can occur when I have two connections to the Internet and two firewalls. Packets may leave through firewall A, but return through firewall B. In that case, firewall A sees the SYN packet go out and will monitor that state of that connection. But when firewall B sees the SYN-ACK reply packet come in form the outside, it has nothing to match it against. It doesn't know about the packet that firewall A saw, so it treats the SYN-ACK packet as a protocol violation and drops it.
In the 2nd solution, all packets pass through the firewall, so it can correctly monitor the state of each connection.